[/quote]
,
![\[ \int_{-1}^{1} e^x \sin \sqrt{1-x^2} \, \mathrm dx .\]](http://latex.artofproblemsolving.com/0/f/0/0f01e4820da5328db8b1b55947de2148cbdd3e83.png)
be a positive semi-definite matrix. Prove that the matrix 
,
.
be an acute triangle with incenter 
and circumcenter 
.
and 
at 
and 
respectively, and 
is the reflection of 
over 
and 
meet at 
,
and 
,
is the midpoint of 
.
and 
,
.
with right angles at 
has an inscribed circle with center 
.
and 
be 
and 
,
and 
be 
.
be a point on the Neuberg cubic. Show that as 
increasing with
?
?
such that 
for all real numbers 
is the greatest integer less than or equal to 
)
be the circumcircle of a non-isosceles triangle 
be a tangent line to 
)
be midpoint of 
and 
.
and 
intersect at points 
.
and 
intersects 
,
intersects 
.
,
.
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
and 
are the angle bisectors of angles 
and 
respectively. Let 
past 
past 
.
at 
.
on 
and 
on 
is cyclic, then 
is tangent to the circumcircle of 
.
Y by RedFlame2112, Adventure10
The 
edges of a regular icosahedron are distinguished by labeling them 
How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?
edges of a regular icosahedron are distinguished by labeling them 
How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?
.
,
given by sending a vector of edge labels to the vector listing the sum of the edges around each face. We want the size of 
.
is surjective: labeling the five edges around a corner 
,
.
.
.
ways to label the lateral edges of the antiprism, and 
ways to label the remaining edges of the antiprism.
for 
over 
are constants. It is easy to verify this has a unique solution, giving a total count of 
.
ways.
.
ways to color the edges of these rhombi, and given such a coloring, each of the remaining 10 diagonals has one or two possible colors. It can be shown that the expected number of ways to color each of these diagonals is 
,
.
,
is the complex number of the color of the 
t
.
are all the same or all different for any edge formed by 
,
.
,
,
,
faces.
,
possible combinations of 
.
(it is easy to check that any term will have some 
,
,
.
.
is a natural choice, but it doesn't quite work, because it can be 
or 
.
makes the polynomial the same for any valid case.
with 4 nodes. Yet how was this gonna help?
.
for the 
of the edges of 
,
.
.
,
is 
,
.
where 
to get 
have nonzero contribution to the sum.
on a face formed by 
,
on the other face. The odd cycle (see the diagram on exponents picked) gives a contradiction.
